Categorical semantics - Revision history

Pierre-Marie Pédrot: /* Compact closed categories */

2011-10-03T23:29:22Z

‎Compact closed categories

2011-10-03T23:26:17Z

‎Compact closed categories

2011-10-03T23:06:01Z

introductory paragraph

2009-03-25T15:24:23Z

‎Modeling ILL

2009-03-25T15:12:55Z

‎Compact closed categories

2009-03-25T14:52:59Z

‎Compact closed categories

2009-03-25T14:48:35Z

‎Compact closed categories

2009-03-25T14:46:06Z

‎Compact closed categories

2009-03-25T14:42:56Z

‎Compact closed categories

2009-03-25T14:40:53Z

‎Compact closed categories

@@ Line 311: / Line 311: @@
 }}
-{{Remark|The above isomorphism does not hold in *-autonomous categories in general.}}
+{{Remark|The above isomorphism does not hold in *-autonomous categories in general. This means that models which are compact closed categories identify <math>\otimes</math> and <math>\parr</math> as well as <math>1</math> and <math>\bot</math>.}}
 {{Proof|

@@ Line 305: / Line 305: @@
 </math>
 It is easy to show that <math>\llcorner \ulcorner f\urcorner\lrcorner=f</math> and <math>\ulcorner\llcorner g\lrcorner\urcorner=g</math> from which we deduce the required bijection.
 }}

@@ Line 20: / Line 20: @@
 {{Definition|title=Monad|
 }}
 == Modeling [[IMLL]] ==

@@ Line 155: / Line 155: @@
 :<math>
 \xymatrix{
-(\mathcal{M},\times,\top)\ar[rr]^{(L,l)}&\bot&\ar[ll]^{(M,m)}(\mathcal{L},\otimes,I)
+(\mathcal{M},\times,\top)\ar@/^/[rr]^{(L,l)}&\bot&\ar@/^/[ll]^{(M,m)}(\mathcal{L},\otimes,I)
+}
 </math>

@@ Line 269: / Line 269: @@
 {{Definition|title=Compact closed category|
 A symmetric monoidal category <math>(\mathcal{C},\tens,I)</math> is ''compact closed'' when every object <math>A</math> has a right dual <math>A^*</math>. We write
-:<math>\eta_A:I\to A^*\tens A</math> and <math>\varepsilon:A\tens A^*\to I</math>
+:<math>\eta_A:I\to A^*\tens A</math> and <math>\varepsilon_A:A\tens A^*\to I</math>
 for the corresponding duality morphisms.
 }}
@@ Line 280: / Line 280: @@
 A compact closed category <math>\mathcal{C}</math> is monoidal closed, with closure defined by
 :<math>\mathcal{C}(A\tens B,C)\cong\mathcal{C}(B,A^*\tens C)</math>
 }}

@@ Line 240: / Line 240: @@
 === Compact closed categories ===
 {{Definition|title=Dual objects|
@@ Line 269: / Line 265: @@
 {{Lemma|
 Two left (resp. right) duals of a same object <math>B</math> are necessarily isomorphic.
 }}

@@ Line 276: / Line 276: @@
 {{Property|
-A compact closed category is monoidal closed.
+A compact closed category <math>\mathcal{C}</math> is monoidal closed, with closure defined by
 }}

@@ Line 264: / Line 264: @@
+}
 </math>
-commute.
+commute. The object <math>A</math> is called a left dual of <math>B</math> (and conversely <math>B</math> is a right dual of <math>A</math>).
 }}
 In a compact closed category the left and right duals of an object <math>A</math> are isomorphic.
 {{Property|

@@ Line 254: / Line 254: @@
 A\tens I\ar[ur]^{A\tens\eta}&&&I\tens A\ar[d]^{\lambda_A}\\
 A\ar[u]^{\rho_A^{-1}}\ar@{=}[rrr]&&&A\\
+}
 </math>